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Stable maps from surfaces to the plane with prescribed branching data.
Autor/es: Mª Carmen Romero Fuster, Catarina Mendes de Jesus, Derek Hacon
Revista: Topology and its Applications
We consider the problem of constructing stable maps from surfaces to the plane with apparent contour given by a set of curves immersed (except possibly with isolated cusps) in the plane. Various constructions are used (1) piecing together regions immersed in the plane (2) modifying an existing stable map by a sequence of codimension one transitions (swallowtails etc) or by surgeries. In (1) the way the regions are pieced together is described by a bipartite graph (an edge C* corresponds to a branch curve C with the vertices of C* corresponding to the two regions containing C). We show that any bipartite graph may be realized by a stable map and we consider the question of realizing graphs by fold maps (i.e. maps without cusps). By using Arnol'd's classification of immersed curves, we list all branch sets with at most two branch curves and four double points realizable by planar fold maps of the torus.